Advertisers are hoping for emergence of legitimate third player to help keep prices in check

Facebook Inc. and Alphabet Inc.’s Google tower so far above the rest of the digital advertising world that no company can claim the mantle of No. 3. But many are trying.

Snapchat parent Snap Inc. believes it has the young eyeballs advertisers crave. Amazon.com Inc., ever the disrupter, has the ability to upend the whole business with its extensive data. Verizon Communications Inc. is betting on its blend of web content, location...